Dear Yosef,
Right. If you start with a family of (d-1)-subsets of a set of size n-1, and add a new element to the ground set and to every set in the family, you reach a new family with parameters n and d with precisely the same graph. I am not aware of general inequalities of this form which makes D grows substantially quicker than what follows from the inequality you have stated.

Indeed moving to complements is interesting. The connectivity condition is not preserved but is replaced by the condition that the induced family on every subset M of N, satisfies the condition that is connected. If we require connectivity both for families and for families then the maximum diameter (for ) is .